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Hindawi Publishing Corporation
EURASIP Journal on Advances in Signal Processing
Volume 2011, Article ID 743916, 13 pages
doi:10.1155/2011/743916

Research Article
Multimode Transmission in Network MIMO Downlink with
Incomplete CSI
Nima Seifi,1 Mats Viberg (EURASIP Member),1 Robert W. Heath Jr.,2
Jun Zhang,3 and Mikael Coldrey4
1 Department

of Signals and Systems, Chalmers University of Technology, 412 96 Gothenburg, Sweden
Wireless Networking and Communications Group (WNCG), Department of Electrical and Computer Engineering,
The University of Texas at Austin, Austin, TX 78712-0240, USA
3
Deptartment of Electronic and Computer Engineering, Hong Kong University of Science and Technology, (HKUST),
Clear Water Bay, Kowloon, Hong Kong
4
Ericsson Research, Ericsson AB, 417 56 Gothenburg, Sweden
2

Correspondence should be addressed to Nima Seifi, nima.seifi@chalmers.se
Received 2 June 2010; Accepted 16 October 2010
Academic Editor: Francesco Verde
Copyright © 2011 Nima Seifi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a cooperative multicell MIMO (a.k.a network MIMO) downlink system with multiantenna base stations (BSs), which
are connected to a central unit and communicate with multiantenna users. In such a network, obtaining perfect channel state
information (CSI) of all users at the central unit to exploit opportunistic scheduling requires a substantial amount of feedback
and backhaul signaling. We propose a scheduling algorithm based only on the knowledge of the average SNR at each user

from all the cooperating BSs, denoted as incomplete CSI. Multimode transmission is applied that is able to adaptively adjust
the number of data streams transmitted to each user. Utilizing the results of random matrix theory, an analytical framework is
proposed to approximate the ergodic rate of each user with different number of data streams. Using these ergodic rates, a joint
user and mode selection algorithm is proposed, where only the scheduled users need to feed back instantaneous CSI. Simulation
results demonstrate that the developed analytical framework provides a good approximation for a practical number of antennas.
While substantially reducing the feedback overhead, it is shown that the proposed scheduling algorithm performs closely to the
opportunistic scheduling algorithm that requires instantaneous CSI feedback from all users.

1. Introduction
Recently, cooperative multicell transmission (also called
network MIMO) has been proposed as an efficient way to
suppress the intercell interference and increase the downlink
capacity of cellular systems [1–3]. In one way of realizing
a network MIMO system, multiple base stations (BSs) are
connected to a central unit via backhaul links. The central
unit coordinates BSs and performs joint scheduling and
signal processing operations. Assuming no limitations with
regards to the capacity, error, and delay in the backhaul, and
upon the availability of perfect channel state information
(CSI) of all users at the central unit, the network MIMO
system in the downlink is equivalent to a MIMO broadcast
channel with per-BS power constraint (PBPC) [4–7].

Many of the previous studies on network MIMO assume
full coordination over the whole system, which is not practical (if not impossible). First, the backhaul links connecting
the BSs and the central unit are subject to transmission
error [8] and delay [9]. They also have limited capacity
which confines the amount of data and CSI sharing [10–
12]. Second, connecting a large number of BSs for joint
processing is of high complexity, which has motivated the

development of coordination strategies at a local scale
[13, 14]. Third, obtaining perfect CSI from all the users
at the central unit, which is indispensable to achieve the
full diversity or multiplexing gains, results in a substantial
training and feedback overhead [15–17].
In this paper, we consider local BS cooperation and focus
on the third limitation, that is, the substantial overhead to


2
obtain the CSI for each active user at the central unit. To
this end, we assume the backhaul links to be perfect and
leave the study of the effect of imperfect backhaul to future
work. We propose a framework that enables scheduling based
only on the knowledge of the average received SNR at each
user from all the cooperating BSs, denoted as incomplete CSI.
This reduces the overhead both on the feedback channel
and the backhaul, since only the selected users (usually a
small number) have to feedback their instantaneous CSI for
precoder design.
1.1. Related Work. The scheduling problem in the singlecell multiuser MIMO downlink has been widely investigated
under various precoding and beamforming strategies [18–
22]. The total number of transmitted data streams in such
systems is upper bounded by the number of BS antennas
under a linear precoding framework. Therefore, if the total
number of receive antennas in the system is greater than
the total number of transmit antennas, the scheduling
will consist of selecting both users and the number of
data streams or modes (note that the term “mode” used
in this paper denotes the number of data streams for a

given user rather than the number of active users, as in
[22] or different MIMO transmission techniques, such as
spatial multiplexing/diversity mode [23]) to each user. Such
multimode transmission improves performance by allowing
a dynamic allocation of the transmission resources among
the users [24]. User and mode selection in a network MIMO
system is more challenging than in its single-cell counterpart.
The increased number of users and BSs in the network
MIMO makes the CSI requirement daunting.
Acquiring CSI at the central unit is one of the limiting
factors for a practical network MIMO system. The availability of perfect CSI, however, has a cardinal role in exploiting
the spatial degrees of freedom in such systems [13]. In
practice, CSI for the downlink is obtained through some
form of training and feedback. In time-division duplexing
(TDD) systems, the CSI is obtained at each BS using the
channel reciprocity (see, e.g., [25]). In frequency-division
duplexing (FDD) systems, since uplink and downlink take
place in widely separated frequency bands, the downlink CSI
is fed back via some explicit feedback channels (see e.g.,
[26]). This places a significant burden on the uplink feedback
channel. The feedback overhead increases with the number
of BSs, users, antennas, and subcarriers and can easily occupy
the whole uplink resources. Furthermore, in both TDD and
FDD systems, the CSI should be forwarded from the BSs to
the central unit which limits the backhaul resources for data
transmission.
The tradeoff between the resources dedicated to CSI
overhead and data transmission in the backhaul has been
recently studied in [15–17], where several multicell system
architectures were compared. It was further shown that the

downlink performance of network MIMO systems is mainly
limited by the inevitable acquisition of CSI rather than by
limited backhaul capacity.
As a solution to this limitation, some authors [27, 28]
have proposed strategies based on local CSI at the BSs

EURASIP Journal on Advances in Signal Processing
and statistical CSI at the central unit, whereas others [29, 30]
consider to serve only certain subsets of users with multiple
BSs. In [31], a decentralized cooperation framework has been
proposed in which all the necessary processing is performed
in a truly distributed manner among the BSs without the
need of any CSI exchange with the central unit. Several BS
cooperation strategies have been studied which consider the
combination of limited-capacity backhaul and imperfect CSI
[32, 33].
1.2. Contributions. In this paper, we develop a scheduling
algorithm for a network MIMO system with multiantenna
users. To reduce the feedback overhead, we adopt a twostep scheduling process: the first step is joint user and mode
selection, and the second step is the feedback and precoder
design which only involves the users selected in the first
step. The two-step multimode transmission strategy was also
proposed in the single-cell MIMO downlink in [22], with
single-antenna users and imperfect CSI at the BS. The main
contributions are as follows.
Ergodic Rate Analysis. We propose an analytical framework
to compute an accurate approximation for the ergodic rate of
each user with different number of data streams, based only
on the knowledge of incomplete CSI for any given location.
Essentially, the aggregate channel from multiple distributed

cooperating BSs can be well approximated as coming from a
single super BS. This enables an efficient method to evaluate
the performance of network MIMO systems without the
need for extensive and computationally intensive MonteCarlo simulations.
Joint User and Mode Selection Algorithm. We use the derived
ergodic user rates as a metric to perform user and mode
selection, which is suitable for the data application without
stringent delay constraint. Since the ergodic rate for each user
is obtained only based on incomplete CSI, the small-scale
fading is not exploited in the proposed strategy. Therefore,
it does not provide small-scale multiuser diversity gain, but
instead, it omits the need for feedback of instantaneous
CSI from a large number of users for scheduling by more
than 93%. It is also shown that the performance of the
proposed user and mode-selection strategy is very close to
the opportunistic scheduling based on instantaneous CSI
feedback from all the users.
1.3. Organization. The rest of the paper is organized as
follows: the system model and transmission strategy are
described in Section 2. In Section 3, some mathematical
preliminaries, which are useful throughout the paper, are
presented. An analytical framework to derive an approximation for the ergodic rate of each user at different modes
is proposed in Section 4. A greedy joint user and mode
selection algorithm based on the derived ergodic rates of
each user is described in Section 5. The performance of the
proposed user and mode-selection algorithm is evaluated
in Section 6. Finally, Section 7 concludes the paper and
discusses the future work.

EURASIP Journal on Advances in Signal Processing
1.4. Notation. Scalars are denoted by lowercase letters,
vectors denoted by boldface lowercase letters, and matrices
denoted by boldface uppercase letters. (·) , (·)∗ , det(·),
log(·), and tr(·) denote transpose, complex conjugate transpose, determinant, 2-base logarithm, and trace of a matrix,
respectively; U(m, n) is the collection of m × n unitary
matrices with unit-norm orthogonal columns. E[·] is the
statistical expectation. [Φ]:(m:n) denotes the matrix obtained
by choosing n − m + 1 columns from Φ starting from the
mth column. [Φ](m,n) denotes the m × n upper-left corner
of a square matrix Φ. λi (Φ) and λmin (Φ) denote the ith
ordered and the smallest eigenvalue of ΦΦ∗ , respectively.
x denotes the Euclidean norm of a complex vector x, and
|S | is the cardinality of a set S; dim(·) is the dimensionality
operator. Further, · denotes the floor operation, Φ ⊗ Ψ
denotes the Kronecker product of the two matrices Φ and
n
Ψ, and Cm denotes the combination of n choosing m.

2. System Model
2.1. Network MIMO Structure. The network MIMO system
considered in this paper comprises B cells, each of which has
a BS with Nt antennas and Kb users, each equipped with
Nr antennas, for Kb ≥ 1 and b = 1, 2, . . . , B. The total
number of active users in the system is denoted as K =
B
b=1 Kb . Users in different locations of the cellular coverage
are subject to distance-dependent pathloss and shadowing. A
narrowband frequency-flat fading channel is considered. We
consider the downlink transmission. The following are the

key assumptions made in this paper.
Assumption 1. All the B cooperating BSs are interconnected
via a central unit with the use of backhaul links with infinite
capacity such that they can fully share CSI and user data.
With this assumption, all the cooperating BSs form a
distributed antenna array that can perform joint scheduling
and transmission.
Assumption 2. The number of antennas at each BS is greater
than that of each user, that is, Nt ≥ Nr .
Due to space constraints, user terminals can only have
a small number of antennas, which makes Nt ≥ Nr a
reasonable assumption. Therefore, each user can have at
most Nr data streams.
Assumption 3. For the scheduling phase, the central unit
relies on the knowledge of incomplete CSI of all users
(scheduling CSI). For the transmission phase, however, the
central unit has perfect knowledge of the singular values
and the corresponding right singular vectors of the selected
eigenmodes of each selected user (transmission CSI) for
precoding design.
This assumption of scheduling CSI significantly reduces
the feedback and backhaul signaling overhead for scheduling.
The transmission CSI assumption is due to the transmission strategy employed in this paper (see Section 2.3),

3
which reduces the transmission CSI required for precoding
design with respect to the strategies which requires the
complete channel matrix of each selected user. The transmission CSI can be reduced even more using limited feedback
techniques [34–36], which we will not explore in this paper.
2.2. Received Signal Model. The aggregate channel matrix of

user k from all the B cooperating BSs can be written as
Hk =

√

ρk,1 Hk,1 · · ·

√

ρk,B Hk,B ,

(1)

where Hk,b ∈ CNr ×Nt represents the small-scale fading
channel matrix and ρk,b is the large-scale fading channel
coefficient that captures the distance-dependent pathloss
including shadowing for user k from the bth BS.
We denote the Nt × 1 transmit signal vector from the bth
BS as xb . Therefore, the BNt × 1 aggregate transmit signal
vector from all the B cooperating BSs can be written as
x = x1 · · · xB

.

(2)

The discrete-time complex baseband signal received by the
kth user is given by
yk = H k x + n k ,

(3)

where nk is the noise vector at the kth user, with entries
that are independent and identically distributed complex
Gaussian with zero mean and unit variance, denoted as i.i.d
CN (0, 1).
2.3. Transmission Strategy. To simultaneously transmit multiple spatially multiplexed streams to multiple users, we
adopt a linear precoding strategy called multiuser eigenmode
transmission (MET) (the framework developed in this paper,
however, can be used with any other linear precoding in
which the precoding matrix for each user is dependent
only on the other users’ channels) [37]. The MET approach
enables the number of data streams for each user to be adaptively selected and at the same time avoids the complexity
of joint iterative precoder/equalizer design [38]. Denote K
as the set of served users at a given time interval and assign
indices k = 1, . . . , |K |. Denote Lk as the set of eigenmodes
selected for transmission to user k, which are indexed from 1
to k , where k = |Lk |. Under a linear precoding framework,
the total number of data streams in the downlink, denoted
as the system transmission mode (STM), is upper bounded by
the number of transmit antennas and can be written as
|K |

L=

j,

(4)

j =1

where 1 ≤ L ≤ BNt . The aggregate transmitted signal is given
by
|K |

x=

Tk dk ,
k=1

(5)


4

EURASIP Journal on Advances in Signal Processing

where Tk ∈ CBNt × k is the precoding matrix and dk denotes
the k dimensional signal vector for user k. It is assumed that
each user k at a given time slot is able to perfectly estimate
its channel matrix Hk without any error. Furthermore, each
user k performs a singular value decomposition (SVD) on its
channel as Hk = Uk Σk Vk . We denote the ith singular value
and the corresponding left and right singular vectors of Hk
as σk,i , uk,i , and vk,i , respectively. We also assume that the
singular values in Σk are arranged in the descending order;
that is, σk,1 ≥ σk,2 ≥ · · · ≥ σk,Nr . The kth user’s receiver is
a linear equalizer given by [Uk ]∗ k ) . Using (3) and (5), the
:(1:
postprocessed signal rk after applying the linear equalizer is

given by

Definition 1. Let Z denote a q × p complex matrix with q ≤
p and a common covariance matrix C = E{z j z∗ } for all j,
j
where z j is the jth column vector of Z. The elements of two
columns zi and z j are assumed to be mutually independent. If
the elements of Z are identically distributed as CN (0, 1) such
that E{Z} = 0, then the Hermitian matrix ZZ∗ is a central
Wishart matrix with p degrees of freedom and covariance
matrix C, denoted as ZZ∗ ∼ CWq (p, C).
3.1. Approximation of a Linear Combination of Wishart
Matrices. Let Ys ∼ CWq (ps , Cs ) for s = 1, 2, . . . , S be
mutually independent central Wishart matrices. Consider a
linear combination
S

|K |

rk = Fk Tk dk + Fk

T j d j + wk ,

(6)

j =1, j = k
/
∗

where Fk = [Uk ]:(1: k ) Hk and wk is the processed noise, which

is still white since the equalizer is a unitary matrix. In the case
of perfect knowledge of F1 , . . . , F|K | , denote the aggregate
interference matrix as Hk = [F∗ · · · F∗−1 F∗ · · · F∗ | ]∗ .
1
|K
k
k+1
To suppress the interuser interference, the constraint F j Tk =
0 must be satisfied for k = j [39]. This requires that Tk lies
/
in the null space of Hk . With this constraint satisfied, the
second term on the right hand side of the equality in (6)
becomes zero. Denote the total number of interfering data
streams for user k from the other (|K | − 1) selected users
|
as k = |jK1, j = k j . As a result, there are only BNt − k
=
/
spatial degrees of freedom available at the transmitter side
to support spatial multiplexing for user k, and therefore,
k ≤ min{Nr , BNt − k }. In [40], it was shown that the
precoding matrix Tk can be written as a cascade of two
precoding matrices Bk and Dk , that is, Tk = Bk Dk , where the
BNt ×(BNt − k ) matrix Bk removes the interuser interference.
Denote the SVD of Hk as
Hk = Uk Σk V(1)
k

V(0)
k

∗

,

(7)

V(0)
k

corresponds to the right singular vectors of Hk
where
associated with the null modes. One natural choice is Bk =
V(0) . As a matter of fact, Fk Bk is the effective interuser
k
interference-free channel for user k. The (BNt − k ) × k
matrix Dk is used for parallelization. Denote the SVD of the
effective channel for user k as
(1)

Fk Bk = Uk Σk Vk

(0) ∗

Vk

,

Y=

(8)

(1)

where Vk denotes the right singular vectors of Fk Bk
corresponding to the first k nonzero singular values. The
(1)
optimum choice of Dk is then Dk = Vk [41].

3. Mathematical Preliminaries
In this section, we present some mathematical preliminaries
from matrix variate distributions and random matrix theory
which prove useful in the analysis to follow. For more
detailed discussions, the readers are referred to [42–44].

αs Ys ,

αs > 0.

(9)

s=1

The distribution of Y can be approximated by the distribution of another Wishart matrix as Y ∼ CWq ( p, C) [42, page
124], where p is the equivalent degrees of freedom given by
⎡

(2q)
S
s=1 αs ps Cs

⎢ det
p=⎣
det

S
2
s=1 αs ps (Cs

⊗ Cs )

⎤1/q2
⎥
⎦

,

(10)

and C denotes the equivalent covariance matrix written as
S

C=

1
αs ps Cs .
p s=1

(11)

Now, if p1 = · · · = pS = p and C1 = · · · = CS = C, then
(10) can be rewritten as
⎡
⎢ p

p=⎣

p

⎤1/q
2q2
S
det (C)(2q) ⎥
s=1 αs
⎦
q2
S
2
det(C ⊗ C)
s=1 αs

2

.

(12)

Using the determinant property of the Kronecker product
[42, Chapter 3], that is, det(C ⊗ C) = det(C)2q , in (12), we
can obtain

⎡

⎢
p = p⎣

2⎤
S
s=1 αs
⎥
⎦.
S
2
s=1 αs

(13)

By substituting (13) in (11), it then holds that
C=

S
2
s=1 αs
S
s=1 αs

C.

(14)

Finally, recall from Definition 1 that the condition q ≤ p

should hold for the Wishart distribution CWq ( p, C) to be
meaningful. In the following theorem, the upper and lower
bound for p are obtained.
Theorem 1. Assume that αs ≥ 0 for s = 1, . . . , S and also that
at least one of the αs ’s is nonzero. If p is defined in (13), then
p ≤ p ≤ Sp. Furthermore, the upper bound equality happens
when α1 = α2 = · · · = αS , while the lower bound equality
holds when ∃!s : αs > 0 (∃! means there exists one and only
one).
Proof. See Appendix A.


EURASIP Journal on Advances in Signal Processing

5

3.2. Truncation of Random Unitary Matrices and Jacobi
Ensemble
Definition 2. If X ∼ CWm (n1 , C) and Y ∼ CWm (n2 , C) are
independent complex Wishart matrices, then J = X(X + Y)−1
is called a complex Jacobi matrix.
It is shown in [45, Proposition 4.1] that J has the same
distribution as that of [U](q,p) [U]∗ , where U ∈ U(n, n)
(q,p)
with q = m, p = n1 , and n = n1 + n2 . Therefore, the
eigenvalues of J are the same as those of [U](q,p) [U]∗ . The
(q,p)
distribution of the extreme eigenvalues of the complex Jacobi
ensemble is derived in [44].

4. Ergodic Rate Analysis
In this section, we derive an approximation for the ergodic
rate of each user k at different modes k . To assist the analysis,
we assume that the elements of Hk,b are distributed such that
Hk,b H∗ ∼ CWNr (Nt , C) for b = 1, . . . , B. Let the precoding
k,b
matrix for user k be written as
Tk = Tk,1 · · · Tk,B

,

(15)

where Tk,b denote the precoding applied at the bth BS for
user k, such that the transmitted signal from the bth BS can
|
be written as xb = |K1 Tk,b dk . Assuming MET and the
k=
practical per-BS power constraint (PBPC) with STM equal
to L, the ergodic rate of a user k with k data streams using
(6) can be expressed as
Rk ( k , L) = E max log det I k + Fk Tk Qk T∗ F∗
k k
Qk

,

Lemma 1. Assume k = L− k , where L is the STM and k is the
mode of user k. Furthermore, assume Bk ∈ U(BNt , (BNt − k ))
is the matrix that projects the channel of user k onto the null

space of other users and is independent of Fk . Assume Fk and
Bk have SVDs given by Fk = UFk ΣFk V∗k and Bk = UBk ΣBk V∗k ,
F
B
respectively. It then holds that

⎡

⎡

k

BP
U k Σk
L

:(1: k )

Σk

∗

∗

:(1: k ) Uk

⎤

BP
= E⎣ log 1 +

λi (Fk Bk ) ⎦,
L
i=1

(a)

(18)
where (a) follows using the matrix identity det(I + AB) =
det(I + BA). Therefore, the ergodic rate of a user k at mode k
depends on the distributions of λi (Fk Bk ) for all i. In order to
compute the distribution of λi (Fk Bk ) in the network MIMO
case, we provide the following result.

0

k

1

f(λi (Fk ),λmin ) (λ, λ )dλ dλ

∞

0

log(λ) f(λi (Fk ),λmin ) (λ, λ )dλ dλ

∞

0

log(λ ) f(λi (Fk ),λmin ) (λ, λ )dλ dλ

k
BP
+
L
i=1

log
k

+

1

i=1 0

(a)

BP
f(λi (Fk ),λmin ) (λ, λ )dλ dλ
L

0

i=1 0
k

BP

λλ
L

∞

1

i=1 0

+

log

log

i=1 0

(17)

=

∞

1

k

=

∗

Rk ( k , L) = E log det I k +

(19)

⎤

k

k

=

subject to

where Qk = E[dk dk ] is the power allocation matrix for user
k and P is the power constraint at each BS. Since the total
power constraint (TPC) over all the BSs is less restrictive, the
performance under TPC is equal or better than that under
PBPC. It has also been shown that there is only a marginal
rate loss of PBPC to TPC [13]. Therefore, for simplicity
and analytical tractability, we assume TPC and equal power
allocation among all the L data streams in the downlink, that
is, Qk = (BP/L)I k . The ergodic rate in (16), can be written
as [37]

∀i.

BP
λi (Fk )λmin ⎦

Rk ( k , L) ≈ E⎣ log
L
i=1

(16)

for b = 1, . . . , B,

,

We denote λmin ([V∗k UBk ](i,(BNt − k )) ) with λmin hereafter
F
in the paper for the ease of notation. Denote the joint
probability density function (pdf) of λi (Fk ) and λmin as
f(λi (Fk ),λmin ) (λ, λ ), and let f(λi (Hk )) (λ) and f(λmin ) (λ ) denote the
marginal pdf of λi (Fk ) and λmin , respectively. Using the result
of Lemma 1 in (18) and the approximation log(1 + x) ≈
log(x), we can get an approximation for the ergodic rate for
user k as

k

≤ P,

(i,(BNt − k ))

Proof. See Appendix B.

+

tr E Tk,b Qk T∗
k,b

V∗k UBk
F

λi (Fk Bk ) ≥ λi (Fk )λmin

∞

0

log(λ) f(λi (Hk )) (λ)dλ

1

i=1 0

log(λ ) f(λmin ) (λ )dλ ,
(20)

where (a) follows from the fact that λi (Fk ) = λi (Hk ) for all i,
which results from (6).
The elements of the aggregate channel matrix Hk in (1)
for any realization of the kth user location do not follow
an i.i.d complex Gaussian distribution in general, due to
the different large-scale channel coefficients to different BSs.
Assuming Hk,m and Hk,n are mutually independent for all
m = n, Hk H∗ = B=1 ρk,b Hk,b H∗ is a linear combination
/

b
k
k,b
of central Wishart matrices, and according to the results in
Section 3, its distribution can be approximated with that
of another Wishart matrix Hk H∗ ∼ WNr (Nt,k , ρk C), where
k
Nt,k and ρk are obtained using (13) and (14) as (since


6

EURASIP Journal on Advances in Signal Processing

the dimensions of Hk must be integers, we use x + 0.5 to
round x to the nearest integer)

Nt,k

⎢
⎢
⎢
⎢
= p⎣

⎥
⎥
⎥
+ 0.5⎥,
⎦

2
B
b=1 ρk,b
B
2
b=1 ρk,b

where Γ(a, b) ¸ b xa−1 e−x dx and γ(a, b) ¸ 0 xa−1 e−x dx
denote the upper and lower incomplete Gamma functions,
(n)(m)
is given by
respectively, αi, j
∞

⎧
⎪i + j − 2
⎪
⎪
⎪
⎨

(21)

and
ρk = ⎝

if i < n and j < m,

(n)(m)

αi, j
= i+ j
⎪

⎪
⎪
⎪
⎩

⎛

b

⎞

if i ≥ n and j ≥ m,

i+ j−1

(26)

otherwise,

and

B
2
b=1 ρk,b ⎠
.
B

b=1 ρk,b

(22)

b−1

ζa,b =

a− j

−1/a−b

.

(27)

j =0

Remark 1. We note that Nt,k is a function of ρk,b for b =
1, . . . , B, which depends on the position of the user k.
Therefore, for user k at any given position in the cell,
the Nr × Nt,k i.i.d channel matrix Hk ∼ CN (0, ρk C) can
be interpreted as if the user is communicating with one
super BS with Nt,k transmit antennas and the equivalent
large-scale channel coefficient ρk . Furthermore, according to
Theorem 1, the maximum of Nt,k is BNt , which corresponds
to positions where ρk,1 = · · · = ρk,B . At other positions,
however, where user k experiences larger ρk,b values from
some of the BSs and smaller from the others, Nt,k will be
smaller than BNt . It can be concluded that Nt,k is determined

mainly by those BSs to which the user has largest ρk,b values,
and those are the ones that help the cooperation and are
actually seen by the user.
Since the distribution of Hk H∗ is approximated with
k
the distribution of another Wishart matrix Hk H∗ , we have
k
fλi (Hk ) (λ) ≈ fλi (Hk ) (λ). The distribution fλi (Hk ) (λ) for i =
1, . . . , Nr for the uncorrelated central case is given in [46] as
(the general framework developed in this paper, however, is
applicable to arbitrarily correlated channels. We only express
the result for the uncorrelated case for simplicity)
Nr

fλucH ) (λ) = Guc
(
i

k

Nr

(−1)n+m λn+m−2+Nt,k −Nr e−λ |Ωuc |,

n=1 m=1

(23)

for i = 1, . . . , Nr ,
where Guc is given by

⎡

Guc = ⎣

Nr

Nr

Nt,k − i !
i=1

⎤−1

Nr − j !⎦ .

(24)

j =1

The (i, j)th element of Ω is written as

ωi, j

⎧
⎪γ α(n)(m) + Nt,k − Nr + 1, λ
⎪
i, j
⎪
⎪
⎨

(n)(m)
= Γ αi, j
+ Nt,k − Nr + 1, λ
⎪
⎪
⎪
⎪
⎩ (n)(m)
αi, j
+ Nt,k − Nr !ζNr ,1

for i = 1,
for i = Nr ,
otherwise,

(25)

In order to find fλmin (λ ), we note that the multiplication
of two unitary matrices is another unitary matrix, that is,
V∗k UBk ∈ U(BNt , BNt ) [47]. Therefore, [V∗k UBk ](i,(BNt − k )) is
F
F
a truncated unitary matrix. As mentioned in Section 3.2, for
any Wishart distributed unitary matrix with Haar measure
A ∈ U(n, n), the multiplication [A](q,p) [A]∗ for q ≤ p ≤
(q,p)
n, has the same distribution as a complex Jacobi ensemble
[45, Proposition 4.1]. The distribution of the minimum
eigenvalues of the complex Jacobi ensemble is obtained in
[44, Equation 3.2]

fλmin (λ ) =

Γi (BNt )Γi (i)
Γi i + k Γi BNt −
× i k λ (BNt −
(1)

×2 F1

k −i−2)

k

(1 − λ )(i k −1)

k − 1, i − BNt + k +2; k +i; (1 − λ

)Ii−1 ,
(28)

where Γm (c) = π m(m−1)/2 m 1 Γ(c − j + 1) denotes the
j=
∞
multivariate Gamma function, Γ(a) ¸ 0 xa−1 e−x dx is the
(1)
Gamma function, and 2 F1 ( k , i − BNt + k ; k + i; (1 − λ )Ii )
is a hypergeometric function of a matrix argument [44, 48].
Based on (23) and (28), we can evaluate (20) numerically.
To verify the accuracy of the approximation in (20), we
consider a hexagonal cellular layout with cell sectoring. By

using 120-degree sectoring in each cell, every 3 neighboring
cells can coordinate with each other to serve users in the
shadow area shown in Figure 1. The number of transmit
antennas is chosen to be Nt = 4, which is the value currently
implemented in wireless standards such as 3GPP LTE, and
Nr = 2. We randomly place two users in each cell sector.
The pathloss model is based on scenario C2 of the WINNER
II specifications [49]. The large-scale fading is modeled as
lognormal with standard deviation of 8 dB. The edge SNR
is defined to be the received SNR at the edge of the cell,
assuming that one BS transmits at full power while all other
BSs are off, accounting for pathloss but ignoring shadowing
and small-scale fading.
Figure 2 depicts the ergodic rate of a sample user k for
k = 1, 2 and different values of L (for the clarity of the plot
we consider only some values of L. The performance for the
other values can be concluded easily). The results correspond


EURASIP Journal on Advances in Signal Processing

7
12

Central
unit

10
8
^

Nt,k

BS 2
6

U2
U3

4

BS 3

2

U1

0
0.2

BS 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized user distance from the home BS

Figure 1: A network MIMO system with 3 BSs, connected to a
central unit via backhaul links.

Figure 3: Equivalent number of transmit antenna Nt,k versus the
normalized distance from the home BS for a sample user k moving
along the line that connects the BS 3 to the center of the shaded
hexagon in Figure 1.

65
60

Sample user rate (bps/Hz)

55
50
45
40
L1

35
30

25
20

L2

15
−5

0

5

10

15

20

Edge SNR (dB)
L = 1, approximation
L = 1, simulation
L = 5, approximation
L = 5, simulation
L = 10, approximation
L = 10, simulation

L = 2, approximation
L = 2, simulation
L = 6, approximation
L = 6, simulation

L = 11, approximation
L = 11, simulation

Figure 2: Ergodic rate of a sample user k versus edge SNR for
a given realization of user locations, different values of L, and
k = 1, 2. The dashed lines indicate the rate obtained from the
lower bound approximation, that is, Rk ( k , L), while the solid lines
represent the rate obtained from the simulation, that is, Rk ( k , L).

to one random snapshot of user locations when for any given
L and k other users are assigned with 1 or 2 data streams. It
is also assumed that no user is within a normalized distance
of 0.2 from its closest BS for the pathloss model to be valid.
It is observed that the lower bound approximation in (20) is
very close to the simulation results obtained by Monte-Carlo
simulations using (16) over the full range of edge SNR. The
difference between the approximation and the achieved rate

is small enough to consider the approximation good enough
for scheduling as explained in the next section.
To justify the argument in Remark 1, in Figure 3, we plot
Nt,k versus the normalized distance from the home BS for a
sample user k moving along the line that connects the BS 3
to the center of the shaded hexagon in Figure 1. It is observed
that within a normalized distance of 0.5 from the BS 3, Nt,k =
4 which results from the fact that ρk,3 is much larger than ρk,1
and ρk,2 , and therefore, only the BS 3 is seen by this user. As
the user moves toward the center of the shaded hexagon, ρk,1
and ρk,2 increase but ρk,3 decreases, resulting an increase in
Nt,k . Indeed at the center of the shaded hexagon Nt,k = 12,

which means all the 3 BSs are seen by the user and actually
can be helpful in the cooperation. Therefore, BS cooperation
is not very helpful for the cell interior user, and only edge
users get most of the benefit. This can be used to design BS
cooperation.

5. Downlink Scheduling: Joint User
and Mode Selection
In this section, downlink scheduling for multimode transmission is discussed. The total number of streams in the
system under study is upper bounded by BNt , and normally
BNt , so at each scheduling phase a subset of users,
KNr
and the preferred mode of each user must be selected for
transmission. In multimode transmission, the number of
data streams for each user is adaptively selected, which allows
to efficiently exploit the available degrees of freedom in the
channel using multimode diversity (multimode diversity is
a form of selection diversity among users with multiple
antennas, which enables the scheduler to perform selection
not only among the users (multiuser diversity), but also
among the different eigenmodes of each user) [24].
In a system with heterogenous users, the goal of the
downlink scheduling is to make the system operate at a rate


8

EURASIP Journal on Advances in Signal Processing

point of its ergodic achievable rate region such that a suitable

concave and increasing network utility function g(·) of the
user individual ergodic rates is maximized [50].
Let M = {1, 2, . . . , KNr } denote the set consisting of all
possible modes for all users, and let Si be a subset of M with
|Si | ≤ BNt . Let Ki denote the set of users with at least one
selected mode in Si . The downlink scheduling problem we
wish to solve is defined as
S = max g
Si

Rk ( k , |Si |)

k

= |Si |,

1≤

gMSRS

Rk ( k , |Si |)

|K |

k∈Ki

k

≤ Nr .

k=1

To solve (29) through brute-force exhaustive search over Si ,
BNt
KNr
m=1 Cm combinations must be checked. Furthermore, for
each combination Si , the knowledge of {Rk ( k , |Si |)}k∈Ki is
required, which is not easy to compute in general. This is a
BNt and very
computationally complex problem if KNr
difficult to implement.
5.1. Low-Complexity User and Mode Selection. To reduce
the computational complexity and at the same time exploit
the benefits of multimode transmission, we propose a lowcomplexity joint user and mode selection algorithm. To
simplify the computation of {Rk ( k , |Si |)}k∈Ki for any given
Si , we propose to use the approximations obtained in (20)
for Rk ( k , |Si |) instead of the exact fomula in (16). This
enables the analytical computation of g({Rk ( k , |Si |)}k∈Ki )
for any given Si , with only the knowledge of the average
SNR of each users to all the BSs, and avoids the complexity
of precoding matrix computations. Therefore, it not only
reduces the computational complexity at the central unit but
also removes the need for any instantaneous CSI feedback
at the expense of sacrificing the small-scale fading multiuser
diversity.
One way to reduce the complexity associated with
exhaustive search is to treat (29) as a relaxed optimization
problem, that is, to greedily select data streams which
maximize the network utility function. Toward this goal, we
gradually increase L from 1 to BNt . For any given L, the

approximate ergodic rate for the next unselected eigenmode
of all users is computed using (20). The algorithm continues
until either L = BNt or the network utility function starts
to decrease. Once the user and mode selection is done, only
selected users need to feedback the singular values and the
corresponding singular vectors of their selected eigenmodes
for precoder design. Therefore, the proposed scheduling
algorithm is of low complexity and is suitable for application
when delay is not a stringent constraint or when the feedback
resources is limited. The resulting algorithm is summarized
in Algorithm 1.
5.2. Network Utility Function. We focus on two special cases
of network utility function, namely, the ergodic sum rate and
the sum log ergodic rate. To perform maximum sum rate

=

1 i
Rk ( k , |Si |).
B k=1

(30)

To introduce fairness by performing proportional fairness
scheduling (PFS) for a given Si , the per-cell sum log ergodic
rate utility function is defined as [50]

,
(29)

|Ki |

subject to

k∈Ki

scheduling (MSRS) for a given Si , the per-cell ergodic sum
rate utility function is defined as

gPFS

Rk ( k , |Si |)

|K |

k∈Ki

=

1 i
log Rk ( k , |Si |) .
B k=1

(31)

6. Simulation Results and Key Observations
In this section, the performance of the proposed user and
mode-selection strategy is evaluated via Monte-Carlo simulations. The assumptions for the cellular layout, pathloss,
shadowing, and the number of antennas are given in
Section 4. We drop K1 = K2 = K3 = 10 users randomly

according to a uniform distribution in each cell.
Inspired from [49], we follow a drop-based simulation.
In this approach, a drop corresponds to one realization
for user locations, during which the large-scale fading
parameters as well as velocity and direction of travel for
users, are practically constant. Therefore, each user can only
undergo small-scale fading at each location. Furthermore,
large-scale fading parameters are realized independently
from drop to drop. This method does not take into account
the time evolution of the channel. The main advantage of
it is the simplicity of the simulation. We run 1000 drops
for user locations. At the beginning of each drop, all the
users feedback their average SNR from all the cooperating
BSs (in real systems, such update is not frequent and only
occurs when users move around) and the set S is obtained
using the Algorithm 1. For each obtained S at each drop,
1000 realizations are simulated with independent small-scale
channel states.
6.1. Sum Rates for Different Systems. Figure 4, compares
the ergodic sum rate of the proposed strategy with both
MSRS and PFS to that of single-user transmission (SUT)
and opportunistic scheduling based on instantaneous CSI
(OSICSI). In SUT, only one user with the best ergodic rate
is selected and served at each scheduling interval. For the
detailed information about the OSICSI algorithm, see [37].
It is shown in Figure 4 that the approximate sum rate is quite
close to the achieved one with MSRS. It can also be observed
that the achieved sum rate for PFS is very close to that of
MSRS. It is further shown that the proposed algorithm for
both MSRS and PFS performs much better than SUT and

achieves a large fraction of the sum rate of OSICSI over a
practical range of edge SNR values. For example, at an edge
SNR of 10 dB, it achieves 80% and 68% of the sum rate of
opportunistic scheduling with MSRS and PFS, respectively.
In Figure 5, the sum log ergodic rate versus edge SNR is
plotted. It is observed that the approximate and simulated
curves are in good agreement.


EURASIP Journal on Advances in Signal Processing

9

(1) Initialization: L = 1, S = ∅, g(∅) = 0
(2) while L ≤ min(KNr , BNt ) do
(3)
r(L) = 0, ν = 0, SL = ∅, KL = ∅, Rk (i, L) = 0, k = 0, uk = 0 for k = 1, . . . , K and i = 1, . . . , Nr
(4)
while ν ≤ L do
(5)
for k = 1 to K do
(6)
Compute sk ← g({R j ( j , L)} j ∈KL−1 ∪ {Rk ( k + 1, L)}) from (20)
j =k
/

(7)
end for
(8)
k max ← arg maxk sk , k max ← k max + 1

(9)
r(L) ← s(k max), uk max ← 1, ν ← ν + 1
(10)
end while
(11)
for k = 1 to K do
(12)
if uk = 0 then
/
(13)
SL = SL ∪ { k } and KL = KL ∪ {k}
(14)
end if
(15)
end for
(16)
if r(L) < r(L − 1) then
(17)
S = SL−1, break
(18)
end if
(19)
L← L+1
(20) end while
Algorithm 1: Pseudocode for the proposed algorithm.

12

Sum log ergodic rate (bps/Hz/cell)

13

45
Ergodic sum rate (bps/Hz/cell)

50

40
35
30
25
20
15
10
5
−5

0

5
10
Edge SNR (dB)

SUT
PFS simulation
MSRS simulation

15

20

11
10
9
8
7
6
5

−5

0

5

10

15

20

Edge SNR (dB)
MSRS approximation
OSICSI
PFS-DMS

Figure 4: Comparison of average sum rate versus edge SNR for
B = 3, K = 30, Nt = 4, and Nr = 2.

To compare the performance of MSRS and PFS, we have

divided the distance between users and their home BS into
bins of width 0.1 of the cell radius, and for each bin, the
fraction of times over all the 1000 drops that a user in that bin
has been scheduled is plotted in Figure 6. It is observed that
with PFS, the activity fraction of users at farther distances
from the BS is higher as compared to that using MSRS.
On the other hand, the simulation results show that the
PFS algorithm almost always chooses 1 data streams for
each served user, while in MSRS 2 data streams are also
selected about 50% of the time that 1 stream is selected.

Approximation
Simulation

Figure 5: Comparison of sum log ergodic rate versus edge SNR for
B = 3, K = 30, Nt = 4, and Nr = 2.

Therefore, in PFS, choosing the users without considering the problem of the mode selection (dominant-mode
transmission) seems to be the relevant strategy, at least
in the chosen scenario. The sum rate of PFS with only
dominant selection (PFS-DMS) is also plotted in Figure 4.
It can be seen that the sum rate of PFS-DMS matches
very well with that of PFS with user and mode selection.
6.2. Feedback Analysis. In this section, we compare the
amount of feedback required by the proposed algorithm with


10

EURASIP Journal on Advances in Signal Processing

BNt , corresponding to their selected eigenmodes at each
small-scale realization for precoding design. If the number of
realizations of small-scale channel states within each drop is
assumed to be T (1000 in this paper), the AFL is given by

25

Activity fraction (%)

20

15

⎡

|K |

⎤

BK
⎦
Fprop =
+ ES ⎣
k (1 + 2BNt ) .
T
k=1

10

5

0
0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized user distance from the home BS

(32)

Since set S changes from drop to drop, the AFL is obtained by
averaging over many drops. For the opportunistic scheduling
based on instantaneous CSI, however, all K users should feed
back their respective Nr × BNt complex channel matrix in all
the T realizations in each drop. Therefore, we have

MSRS
PFS

Fopp = 2BKNt Nr .

Figure 6: Comparison of activity fraction in percentage for both
MSRS and PFS for edge SNR of 10 dB.

AFL per cell

103

102

5

10

15

20

25

30

Number of users per cell

(33)

In Figure 7, the AFL per cell versus the number of users
per cell is compared for the proposed scheduling algorithm
and the opportunistic scheduling based on instantaneous
CSI. It is observed that for the proposed algorithm, the AFL
increases very slowly with the number of users, since no
matter how big the number of users is in the cell, the number
of served users will be limited by the maximum number
of transmit antennas. The only overhead of increasing the
number of users is the average SNR values that should be
fed back at the beginning of each drop, which is negligible
with respect to the amount fed back during a drop. For
the opportunistic scheduling, however, the AFL increases
linearly with the number of users since it has to feed back
the CSI for all the users at each realization. It is observed that
for K1 = K2 = K3 = 30, AFL is decreased by more than 93%
which makes the proposed algorithm attractive in such kind
of scenarios.

Opportunistic scheduling
Proposed algorithm

Figure 7: Comparison of AFL per cell for the proposed algorithm
and the opportunistic scheduling based on instantaneous CSI with
edge SNR of 10 dB.

that by opportunistic scheduling based on instantaneous
CSI. We define the average feedback load (AFL) as the
average number of real coefficients that are fed back to the
central unit during each drop normalized to the number of
small-scale realizations within that drop. For the proposed

algorithm (the results are almost the same for both MSRS
and PFS since the AFL depends on the total number of
transmitted data stream, that is, STM, which is the same on
average for both schemes. We only plot the result for MSRS
here), once the set S is obtained at the beginning of a drop,
each selected user k ∈ K sends back k real-valued singular
values and k complex-valued right singular vectors of size

7. Conclusion
In this paper, we propose an analytical framework to
approximate ergodic rates of users with different modes in
a network MIMO system, based only on the knowledge
of received average SNR from all the cooperating BSs at
each user, called incomplete CSI in this paper. Based on the
derived approximate ergodic rates, the problem of downlink
scheduling with both MSRS and PFS is addressed. The
proposed scheduling algorithm significantly reduces the
feedback amount and performs close to the opportunistic
scheduling based on instantaneous CSI. It is of particular
interest for applications where there is a total feedback
overhead constraint and/or when there is no stringent delay
constraint. It is also shown that by introducing fairness, the
probability of selecting higher modes for each users decreases
significantly, which results in dominant mode transmission
(beamforming) to each user.


EURASIP Journal on Advances in Signal Processing

11

Now, let us define V = v1 (Bk ), v2 (Bk ), . . . , vi (Bk ) , where
v j (Bk ) denotes the jth column of VBk and a1 , a2 , . . . , an is
the span of a1 , a2 , . . . , an . We have

Appendices
A. Proof of Theorem 1
In order to prove the lower bound for p, we write
⎡

λi (Fk Bk ) ≥

2⎤
S
s=1 αs
⎥
⎦
S
α2
s=1 s

⎢

p = p⎣
⎡

S
2
s=1 αs

= p⎣

S
2
s=1 αs
S
2
s=1 αs

(a)

≥ p

(a)

S
S
j =1, j = i
i=1
/
S
α2
s=1 s

+

αi α j

x ∈V
x =1

⎤
⎦

(A.1)

⎝

=

⎞⎛

S

α2 ⎠⎝
s

S

⎝

⎞2

2
βs ⎠.

(c)

min
ΣFk V∗k UBk
F

x ∈V
x =1

=

≥

⎛

(A.2)
(e)

≥

min λi (Fk )
x ∈V
x =1

⎞

S

α2 ⎠.
s

(A.3)

s=1

ΣFk

min
x ∈V
x =1

s=1

αs ⎠ ≤ S⎝

s=1

=1

min
F k U Bk
x ∈V
x =1

(d)

⎞

Now, setting β1 = · · · = βS = 1, (A.2) can be rewritten as
S

2

Fk UBk IBNt ×m x

=

s=1

⎛

min

x ∈V

x

αs βs ⎠ ≤ ⎝

s=1

(b)

= p,

⎛

⎞2

S

2

= min Fk UBk ΣBk x

where (a) follows from the fact that αs > 0 for all

s. The equality for the lower bound holds only when
S
S
i=1
j =1, j = i αi α j = 0, and this happens only when ∃!s :
/
αs > 0. The proof for the upper bound is established using
the Cauchy-Schwarz inequality. For some βs > 0 for all s, we
have
⎛

2

min
Fk Bk x
x∈V
x =1

(f)

:(1:m) x

VFk

:(1,i)

∗

:(1:i)

(i,m)

2

U Bk

:(1:m) x

(i,m)

x

.

The upper bound for p using (A.3) is now obtained as
⎡

p ≤ p⎣

S

S
2
s=1 αs
S
2
s=1 αs

(B.6)

⎤
⎦ = Sp.

(A.4)

The equality condition for the Cauchy-Schwarz inequality
is only satisfied when the vectors α = [α1 · · · αS ] and β =
[β1 · · · βS ] are linearly dependent; that is, α = xβ for some
scalar x. Since in our case β = [1 · · · 1], it results that the
equality holds only when α1 = · · · = αS . This completes the
proof of the Theorem.

Here, (a) follows from the fact that (i) x ∈ V is equivalent
to x ∈ V for some x = V∗k x; and (ii) x 2 = x 2 . (b)
B
follows from the fact that the singular values of a unitary
matrix are all equal to 1. (c) follows from the fact that
UFk y = y , where y = ΣFk V∗k [UBk ]:(1:m) x . (d) follows
F
from the fact that defining z = [UBk ]:(1:m) x , we have
ΣFk V∗k z
F

2

ΣFk

=

+

B. Proof of Lemma 1
First, notice that for values of i > k or i > BNt − k ,
λi (Fk ), or λi (Bk ) are defined as zero. Hence, the argument
of this lemma is obvious for these cases. Now, we prove the
argument for i ≤ k . For notational convenience, we define
m = BNt − k . According to “max-min” half of the CourantFischer Theorem [47], for any subspace M ⊂ CBNt − k with
dimention i, we have
λi (Fk Bk ) =

max

M,dim(M)=i

min
Fk Bk x 2 .
x∈M
x =1

2

2

V∗k UBk
F

V∗k UBk
F

= λi (Fk )λmin

2

:(1:m) x

(B.5)

:(1:i)

ΣFk

VFk

∗

:(1:i) z

1/2
:(i+1:BNt )

VFk

2

∗

:(i+1:BNt ) z

2

(B.7)
.

(e) follows from the fact that with q = [V∗k UBk ](i,m) x , we
F
have [ΣFk ]:(1:i) q 2 ≥ λi (Fk ) q 2 . Finally, (f) follows from
the Rayleigh-Ritz Theorem [47]. Equation (B.6) completes
the proof of the Lemma.

Acknowledgments
The authors would like to thank Professor Plamen Koev
from San Jose State University for the valuable discussions
regarding the distribution of the minimum eigenvalues of the
complex Jacobi ensemble. This work has been supported in
part by VINNOVA within the VINN Excellence Center Chase


12

EURASIP Journal on Advances in Signal Processing

and in part by SSF within the Strategic Research Center
Charmant.
[15]

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